 + Jx i=1N/2 ( c2i - c2i†) ( c2i+1 + c2i+1†),

which is identical to the fermionic expression of the Kitaev spin model in the limit \(J_y=0\)  [1]}. To solve this, we must transform the four-fermion term. We
introduce the Majorana fermions on each sublattice \(\eta _{\circ /\bullet }\)  and \(\gamma _{\circ /\bullet }\)  [2]}, [3]}:
\({\left\lbrace \begin{array}{ll}c_{\circ } = \left( \eta _{\circ } + i\gamma _{\circ } \right)/2, \\c_{\bullet } = \left( \gamma _{\bullet } + i\eta _{\bullet } \right)/2.\end{array}\right.}\) 
